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Problem EExpecting Rain

The Bluewater Geocloud Organization (BGO) has recently
developed brand new software able to predict with pinpoint
precision at which second any particular cloud will start and
stop raining, and with what intensity. There is, however, some
uncertainty about how a cloud will move around; for each zip
code, each cloud will be over that zip code with some
probability.

You have scraped some information about your zip code from
the BGO website, and want to use it to plan your walk to the
bus stop. You wish to minimize the expected amount of rain that
would fall on you. To reach the bus you must get to the bus
stop within $t$ seconds
from now. You have timed your walking speed to be exactly
$1 \frac{m}{s}$.

To complicate matters, some parts of the walk to the bus are
covered by roofs where it might be beneficial to make shorts
breaks whilst waiting for the worst rain to pass. Your front
door (at $d$ meters from
the bus stop) is always under a roof – but the bus stop need
not be.

Input

The first line of input is four space-separated integers:
$d$ ($1 \leq d \leq 1\, 000$), the distance
to the bus stop in meters, $t$ ($d
\leq t \leq 10\, 000$) the time until the bus leaves,
$c$ ($0 \leq c \leq 1\, 000\, 000$), the
number of clouds tracked by BGO, and finally $r$ ($0
\leq r \leq d$), the number of roofs. The next
$c$ lines describe the
clouds; the $i$’th such
line contains four numbers $s_
i$, $e_ i$,
$p_ i$ and $a_ i$ describing the $i$’th cloud:

$s_ i$
($0 \leq s_ i < t$)
is an integer giving the number of seconds until the cloud
starts its raining period,

$e_ i$
($s_ i < e_ i \leq
t$) is an integer giving the number of seconds until
the cloud ends its raining period,

$p_ i$
($0 \leq p_ i \leq 1$)
is a real number (with at most $6$ digits after the decimal
point) giving the probability that the cloud is in your zip
code during its raining period, and

$a_ i$
($0 \leq a_ i \leq
100$) is an integer indicating the amount of rain
the cloud will release during its raining period, given as
nm per second.

Finally $r$ roof
segments follow, each on its own line; the $j$’th such line contains two integers
$x_ j$ and $y_ j$ ($0 \leq x_ j < y_ j \leq d+1$),
indicating that there is a roof segment starting at distance
$x_ j$ away from home,
ending at distance $y_ j$
away from home along the route to the bus stop. Both your home,
the bus stop an the entire route between them are in the same
zip code. No two roofs overlap, however one roof may start at
the same exact location as another ends.

Output

The output consists of single a real value, the minimum
amount of rain in nm you can expect on your route if you reach
the bus stop in time. Answers with absolute or relative
precision $10^{-5}$ of the
actual value will be accepted.